Conditions for proving by mathematical induction to be explanatory

In this paper we consider proving to be the activity in search for a proof, whereby proof is the final product of this activity that meets certain criteria. Although there has been considerable research attention on the functions of proof (e.g., explanation), there has been less explicit attention in the literature on those same functions arising in the proving process. Our aim is to identify conditions for proving by mathematical induction to be explanatory for the prover. To identify such conditions, we analyze videos of undergraduate mathematics students working on specially designed problems. Specifically, we examine the role played by: the problem formulation, students’ experience with the utility of examples in proving, and students’ ability to recognize and apply mathematical induction as an appropriate method in their explorations. We conclude that particular combinations of these aspects make it more likely that proving by induction will be explanatory for the prover.     

Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors

Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory—the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument.     

Gesture in collaborative mathematics problem-solving

In this paper we examine the significance of gestures in a setting where two students try to make sense of, and solve, mathematical problems involving speed and time. We are particularly interested in exploring the claims that gesture serves various signaling functions in collaborative problem-solving communication generally and in mathematics problem-solving more specifically, and that gesture has a diagnostic role for the collaborators and for teachers. The overall purpose of our paper is to illustrate the integral role of gesture in dyadic communication where core problem domain concepts may be difficult to explicate.     

Engaging students in roles of proof

de Villiers (1990) suggested five roles of proof important in the professional mathematics community that may also serve to meaningfully engage students in learning proof: verification, explanation, systematization, discovery, and communication. We investigate written reflections on an end-of-semester assignment from undergraduates in an inquiry-based transition to proof course, where students reflected on instances during the semester when they engaged in the five roles of proof. We present the types of activities students recalled as influential to their engagement in the roles of proof (presenting, discussing, conjecturing, working on problem sets, and critiquing) and describe how students perceived these activities as influential to their engagement in the roles of proof. We provide student quotations highlighting these activities and offer implications for both research and practice.     

How some research mathematicians and statisticians use proof in undergraduate mathematics

The paper examines the roles and purposes of proof mentioned by university research faculty when reflecting on their own teaching and teaching at their institutions. Interview responses from 14 research mathematicians and statisticians who also teach are reported. The results suggest there is a great deal of variation in the role and purpose of proof in and among mathematics courses and that factors such as the course title, audience, and instructor influence this variation. The results also suggest that, for this diverse group, learning how to prove theorems is the most prominent role of proof in upper division undergraduate mathematics courses and that this training is considered preparation for graduate mathematics studies. Absent were responses discussing proof's role in preparing K-12 mathematics teachers. Implications for a proof and proving landscape for school mathematics are discussed.     

Conceptual blending: Student reasoning when proving “conditional implies conditional” statements

Conceptual blending describes how humans condense information, combining it in novel ways. The blending process may create global insight or new detailed connections, but it may also result in a loss of information, causing confusion. In this paper, we describe the proof writing process of a group of four students in a university geometry course proving a statement of the form conditional implies conditional, i.e., (p → q) ⇒ (r → s). We use blending theory to provide insight into three diverse questions relevant for proof writing: (1) Where do key ideas for proofs come from?, (2) How do students structure their proofs and combine those structures with their more intuitive ideas?, and (3) How are students reasoning when they fail to keep track of the implication structure of the statements that they are using? We also use blending theory to describe the evolution of the students’ proof writing process through four episodes each described by a primary blend.     

Derived sequences and reverse mathematics

One of the earliest applications of transfinite numbers is in the construction of derived sequences by Cantor [2]. In [6], the existence of derived sequences for countable closed sets is proved in ATR₀. This existence theorem is an intermediate step in a proof that a statement concerning topological comparability is equivalent to ATR₀. In actuality, the full strength of ATR₀ is used in proving the existence theorem. To show this, we will derive a statement known to be equivalent to ATR₀, using only RCA₀ and the assertion that every countable closed set has a derived sequence. We will use three of the subsystems of second order arithmetic defined by H. Friedman ([3], [4]), which can be roughly characterized by the strength of their set comprehension axioms. RCA₀ includes comprehension for Δ definable sets, ACA₀ includes comprehension for arithmetical sets, and ATR₀ appends to ACA₀ a comprehension scheme for sets defined by transfinite recursion on arithmetical formulas. MSC: 03F35, 54B99.     

Difference and differential equations in the mathematics of finance

Many problems arising in the mathematics of finance involve identical money flows at regular time intervals and are typically solved by appropriate valuation at a focal date or by setting up an equation of value. It is shown here that such problems can be viewed as special cases of a certain class of first‐order difference equations. Problems relating to continuous money flows can be viewed analogously as special cases of a certain class of ordinary first‐order differential equations. Students should thus be encouraged to view the concepts and techniques of the mathematics of finance as not being inherently different from those prevalent in more traditional applied mathematics.     

N.G. de Bruijn’s contribution to the formalization of mathematics

N.G. (Dick) de Bruijn was the first to develop a formal language suitable for the complete expression of a mathematical subject matter. His formalization does not only regard the usual mathematical expressions, but also all sorts of meta-notions such as definitions, substitutions, theorems and even complete proofs. He envisaged (and demonstrated) that a full formalization enables one to check proofs automatically by means of a computer program. He started developing his ideas about a suitable formal language for mathematics in the end of the 1960s, when computers were still in their infancy. De Bruijn was ahead of his time and much of his work only became known to a wider audience much later. In the present paper we highlight de Bruijn’s contributions to the formalization of mathematics, directed towards verification by a computer, by placing these in their time and by relating them to parallel and later developments. We aim to explain some of the more technical aspects of de Bruijn’s work to a wider audience of interested mathematicians and computer scientists.     

Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course

It is widely accepted by mathematics educators and mathematicians that most proof-oriented university mathematics courses are taught in a “definition-theorem-proof” format. However, there are relatively few empirical studies on what takes place during this instruction, why this instruction is used, and how it affects students’ learning. In this paper, I investigate these issues by examining a case study of one professor using this type of instruction in an introductory real analysis course. I first describe the professor’s actions in the classroom and argue that these actions are the result of the professor’s beliefs about mathematics, students, and education, as well as his knowledge of the material being covered. I then illustrate how the professor’s teaching style influenced the way that his students attempted to learn the material. Finally, I discuss the implications that the reported data have on mathematics education research.     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Prior decisions and experiences about mathematics of students in bridging courses

We report on the survey responses of 51 students attending mathematics bridging courses at a major Australian university, investigating what mathematics, if any, these students had studied in the senior years of schooling and what factors affected their decisions about the level of mathematics chosen. Quantitative findings are augmented by qualitative responses to open-ended questions in the survey as well as excerpts from follow-up emails. The findings show that the major reasons for students taking lower levels of mathematics in senior year(s), or dropping mathematics, include finding enough time for non-mathematics subjects, confidence in mathematical capability, advice and maximizing potential ranking for university admission.     

Reinventing the formal definition of limit: The case of Amy and Mike

Relatively little is known about how students come to reason coherently about the formal definition of limit. While some have conjectured how students might think about limits formally, there is insufficient empirical evidence of students making sense of the conventional ?-δ definition. This paper provides a detailed account of a teaching experiment designed to produce such empirical data. In a ten-week teaching experiment, two students, neither of whom had previously seen the conventional ?-δ definition of limit, reinvented a formal definition of limit capturing the intended meaning of the conventional definition. This paper focuses on the evolution of the students’ definition, and serves not only as an existence proof that students can reinvent a coherent definition of limit, but also as an illustration of how students might reason as they reinvent such a definition.     

Examining undergraduate student retention in mathematics using network analysis and relative risk

Higher education faces challenges in retaining students who require a command of numeracy in their chosen field of study. This study applies an innovative combination of relative risk and social network analysis to enrolment data of a single cohort of commencing students from an Australian regional university. Relative risk, often used in epidemiology studies, is used to strategically investigate whether first year mathematics subjects at the university demonstrated a higher risk of attrition when compared to other subjects offered in the first year of study. The network analysis is used to illustrate the connections of those mathematics subjects, identifying service subjects through their multiple connections. The analysis revealed that attrition rates for eight of the nine subjects were within acceptable limits, and this included identified service subjects. The exception highlighted the issue of mathematics competencies in this cohort. This combined analytical technique is proposed as appropriate for use when investigating attrition and retention at faculty and institutional levels, including the determination of levels of intervention and support for any subject.     

Three modes of STEM integration for middle school mathematics teachers

Of the four subjects in an integrated science, technology, engineering, and mathematics (STEM) approach, mathematics has not received enough focus. This could be in part because mathematics teachers may be apprehensive or unsure about how to implement integrated STEM education in their classrooms. There are benefits to integrated STEM in a mathematics classroom though, including increased motivation, interest, and achievement for students. This article discusses three methods that middle school mathematics teachers can utilize to integrate STEM subjects. By focusing on open‐ended problems through engineering design challenges, mathematical modeling, and mathematics integrated with technology middle school students are more likely to see mathematics as relevant and valuable. Important considerations are discussed as well as recent research with these approaches.     

Investigating the association between students' strategy use and mathematics achievement

This observational study used data from 270 second-grade students to investigate the association between students' strategy use for multidigit addition and subtraction and their mathematics achievement. Based on strategies they used during a mathematics interview, students were classified into the following strategy groups: (a) standard algorithm, (b) invented, (c) mixed, and (d) unclassified. We used two-level hierarchical linear regression to investigate the association between students' strategy use and their performance on a standardized test in mathematics. Results indicated that students in the mixed strategy groups had significantly higher mathematics achievement than those in the standard algorithm and the unclassified groups.     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

Seeking mathematics success for college students: a randomized field trial of an adapted approach

Many students enter the Canadian college system with insufficient mathematical ability and leave the system with little improvement. Those students who enter with poor mathematics ability typically take a developmental mathematics course as their first and possibly only mathematics course. The educational experiences that comprise a developmental mathematics course vary widely and are, too often, ineffective at improving students’ ability. This trend is concerning, since low mathematics ability is known to be related to lower rates of success in subsequent courses. To date, little attention has been paid to the selection of an instructional approach to consistently apply across developmental mathematics courses. Prior research suggests that an appropriate instructional method would involve explicit instruction and practising mathematical procedures linked to a mathematical concept. This study reports on a randomized field trial of a developmental mathematics approach at a college in Ontario, Canada. The new approach is an adaptation of the JUMP Math program, an explicit instruction method designed for primary and secondary school curriculae, to the college learning environment. In this study, a subset of courses was assigned to JUMP Math and the remainder was taught in the same style as in the previous years. We found consistent, modest improvement in the JUMP Math sections compared to the non-JUMP sections, after accounting for potential covariates. The findings from this randomized field trial, along with prior research on effective education for developmental mathematics students, suggest that JUMP Math is a promising way to improve college student outcomes.     

Policy and the standards debate: mapping changes in assessment in mathematics

ABSTRACT

The influences on governments for policy changes in schools range across many agencies, including the political party in power. When policies change, the sources of these influences are not always clear. The project whose work is presented in this special issue examines what these changes look like in terms of the differences in assessment tasks of school pupils’ mathematics, over time. In this article we attempt to develop a graph, which we argue will have general applicability internationally, that can help to reveal the sources and nature of those influences. We construct the graph in interaction with an examination of the most recent changes in two countries. We argue that our analysis is a necessary complement to the project’s findings in that it enables us to identify the fields of recontextualisation, their relative strengths in terms of influence and hence conjecture their impact on the mathematics curriculum.     

Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students’ progressively sophisticated ways of reasoning and instructional design heuristics that can facilitate this process. In this article we analyze the case of student reasoning with analytic expressions as they reinvent solutions to systems of two differential equations. The significance of this work is twofold: it includes an elaboration of the Realistic Mathematics Education instructional design heuristic of emergent models to the undergraduate setting in which symbolic expressions play a prominent role, and it offers teachers insight into student thinking by highlighting qualitatively different ways that students reason proportionally in relation to this instructional design heuristic.